Structure dependent sampling in compressed sensing: Theoretical guarantees for tight frames

Many of the applications of compressed sensing have been based on variable density sampling, where certain sections of the sampling coefficients are sampled more densely. Furthermore, it has been observed that these sampling schemes are dependent not only on sparsity but also on the sparsity structure of the underlying signal. This paper extends the result of Adcock, Hansen, Poon and Roman (arXiv:1302.0561, 2013) [2] to the case where the sparsifying system forms a tight frame. By dividing the sampling coefficients into levels, our main result will describe how the amount of subsampling in each level is determined by the local coherences between the sampling and sparsifying operators and the localized level sparsities – the sparsity in each level under the sparsifying operator.     

Iterative hard thresholding for compressed data separation

We study the problem of reconstructing signals’ distinct subcomponents, which are approximately sparse in morphologically different dictionaries, from a small number of linear measurements. We propose an iterative hard thresholding algorithm adapted to dictionaries. We show that under the usual assumptions that the measurement system satisfies a restricted isometry property (adapted to a composed dictionary) condition and the dictionaries satisfy a mutual coherence condition, the algorithm can approximately reconstruct the distinct subcomponents after a fixed number of iterations.     

Deterministic constructions of compressed sensing matrices

Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signal classes is directly related to Gelfand widths. Similar to the deeper constructions of optimal subspaces in Gelfand widths, most sampling algorithms are based on randomization. However, for possible circuit implementation, it is important to understand what can be done with purely deterministic sampling. In this note, we show how to construct sampling matrices using finite fields. One such construction gives cyclic matrices which are interesting for circuit implementation. While the guaranteed performance of these deterministic constructions is not comparable to the random constructions, these matrices have the best known performance for purely deterministic constructions.     

Optimized projections for compressed sensing via rank-constrained nearest correlation matrix

Optimizing the acquisition matrix is useful for compressed sensing of signals that are sparse in overcomplete dictionaries, because the acquisition matrix can be adapted to the particular correlations of the dictionary atoms. In this paper a novel formulation of the optimization problem is proposed, in the form of a rank-constrained nearest correlation matrix problem. Furthermore, improvements for three existing optimization algorithms are introduced, which are shown to be particular instances of the proposed formulation. Simulation results show notable improvements and superior robustness in sparse signal recovery.     

Welch bounds for cross correlation of subspaces and generalizations

Lower bounds on the maximal cross correlation between vectors in a set were first given by Welch and then studied by several others. In this work, this is extended to obtaining lower bounds on the maximal cross correlation between subspaces of a given Hilbert space. Two different notions of cross correlation among spaces have been considered. The study of such bounds is done in terms of fusion frames, including generalized fusion frames. In addition, results on the expectation of the cross correlation among random vectors have been obtained.     

Inversion algorithm based on the generalized objective functional for compressed sensing

Owing to providing a novel insight for signal and image processing, compressed sensing (CS) has attracted increasing attention. The accuracy of the reconstruction algorithms plays an important role in real applications of the CS theory. In this paper, a generalized reconstruction model that simultaneously considers the inaccuracies on the measurement matrix and the measurement data is proposed for CS reconstruction. A generalized objective functional, which integrates the advantages of the least squares (LSs) estimation and the combinational M-estimation, is proposed. An iterative scheme that integrates the merits of the homotopy method and the artificial physics optimization (APO) algorithm is developed for solving the proposed objective functional. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. For the cases simulated in this paper, the reconstruction accuracy is improved, which indicates that the proposed algorithm is successful in solving CS inverse problems.     

Deterministic bounds for restricted isometry in compressed sensing matrices

A new quality measure for compressed sensing matrices is considered and its non-trivial lower bound is obtained which depends on the matrix sizes only. A relation of the new estimate to the standard restricted isometry property is established. The tightness of the new result is discussed as well as some its consequences related to the known results in compressed sensing theory.     

Uniform recovery from subgaussian multi-sensor measurements

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.     

Oversampling and preservation of tightness in affine frames

The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.

     

The High Order Block RIP Condition for Signal Recovery

In this paper, we consider the recovery of block sparse signals, whose nonzero entries appear in blocks (or clusters) rather than spread arbitrarily throughout the signal, from incomplete linear measurements. A high order sufficient condition based on block RIP is obtained to guarantee the stable recovery of all block sparse signals in the presence of noise, and robust recovery when signals are not exactly block sparse via mixed $l_2/l_1$ minimization. Moreover, a concrete example is established to ensure the condition is sharp. The significance of the results presented in this paper lies in the fact that recovery may be possible under more general conditions by exploiting the block structure of the sparsity pattern instead of the conventional sparsity pattern.     

Performance comparisons of greedy algorithms in compressed sensing

Compressed sensing has motivated the development of numerous sparse approximation algorithms designed to return a solution to an underdetermined system of linear equations where the solution has the fewest number of nonzeros possible, referred to as the sparsest solution. In the compressed sensing setting, greedy sparse approximation algorithms have been observed to be both able to recover the sparsest solution for similar problem sizes as other algorithms and to be computationally efficient; however, little theory is known for their average case behavior. We conduct a large‐scale empirical investigation into the behavior of three of the state of the art greedy algorithms: Normalized Iterative Hard Thresholding (NIHT), Hard Thresholding Pursuit (HTP), and CSMPSP. The investigation considers a variety of random classes of linear systems. The regions of the problem size in which each algorithm is able to reliably recover the sparsest solution is accurately determined, and throughout this region, additional performance characteristics are presented. Contrasting the recovery regions and the average computational time for each algorithm, we present algorithm selection maps, which indicate, for each problem size, which algorithm is able to reliably recover the sparsest vector in the least amount of time. Although no algorithm is observed to be uniformly superior, NIHT is observed to have an advantageous balance of large recovery region, absolute recovery time, and robustness of these properties to additive noise across a variety of problem classes. A principle difference between the NIHT and the more sophisticated HTP and CSMPSP is the balance of asymptotic convergence rate against computational cost prior to potential support set updates. The data suggest that NIHT is typically faster than HTP and CSMPSP because of greater flexibility in updating the support that limits unnecessary computation on incorrect support sets. The algorithm selection maps presented here are the first of their kind for compressed sensing. Copyright © 2014 John Wiley & Sons, Ltd.     

基于部分三角函数变换矩阵的块压缩感知测量方法

为了提高块压缩感知的测量效率和重构性能,根据离散余弦变换和离散正弦变换具有汇聚信号能量的特性,提出了基于重复块对角结构的部分离散余弦变换partial discrete cosine transform in repeated block diagonal structure,简称PDCT-RBDS和部分离散正弦变换partial discrete sine transform in repeated block diagonal structure简称PDST-RBDS的两种压缩感知测量方法.所采用的测量矩阵是一种低复杂度的结构化确定性矩阵, 满足受限等距性质.并得到一个与采样能量有关的受限等距常数和精确重构的测量数下限.通过与采用重复块对角结构的部分随机高斯矩阵和部分贝努利矩阵的图像压缩感知对比,结果表明PDCT-RBDS和PDST-RBDS重构的PSNR大约提高1---5dBSSIM提高约0.05, 所需的重构时间和测量矩阵的存储空间大大减少.该方法特别适合大规模图像压缩及实时视频数据处理场合.     

Robustness of orthogonal matching pursuit under restricted isometry property

Orthogonal matching pursuit(OMP)algorithm is an efcient method for the recovery of a sparse signal in compressed sensing,due to its ease implementation and low complexity.In this paper,the robustness of the OMP algorithm under the restricted isometry property(RIP) is presented.It is shown that δK+√KθK,11is sufcient for the OMP algorithm to recover exactly the support of arbitrary K-sparse signal if its nonzero components are large enough for both l2bounded and l∞bounded noises.     

Further results on the connectivity of Parseval frame wavelets

In a previous paper, the authors introduced new ideas to treat the problem of connectivity of Parseval frames. With these ideas it was shown that a large set of Parseval frames is arcwise connected. In this article we exhibit a larger class of Parseval frames for which the arcwise connectivity is true. This larger class fails to include all Parseval frames.

     

Compressed data separation via dual frames based split-analysis with Weibull matrices

In this paper, we consider data separation problem, where the original signal is composed of two distinct subcomponents, via dual frames based Split-analysis approach. We show that the two distinct subcomponents, which are sparse in two diff erent general frames respectively, can be exactly recovered with high probability, when the measurement matrix is a Weibull random matrix （not Gaussian） and the two frames satisfy a mutual coherence property. Our result may be significant for analysing Split-analysis model for data separation.     

CoSaMP: Iterative signal recovery from incomplete and inaccurate samples

Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(Nlog²N), where N is the length of the signal.     

Recovery of signals under the condition on RIC and ROC via prior support information

In this paper, the sufficient condition in terms of the RIC and ROC for the stable and robust recovery of signals in both noiseless and noisy settings was established via weighted $l_1$ minimization when there is partial prior information on support of signals. An improved performance guarantee has been derived. We can obtain a less restricted sufficient condition for signal reconstruction and a tighter recovery error bound under some conditions via weighted $l_1$ minimization. When prior support estimate is at least 50% accurate, the sufficient condition is weaker than the analogous condition by standard $l_1$ minimization method, meanwhile the reconstruction error upper bound is provably to be smaller under additional conditions. Furthermore, the sufficient condition is also proved sharp.